Our experimental set-up and main results are summarized in Fig. REF . Photographs of our acoustic crystal can be seen in Fig. REF (a,b). It consists of identical cylindrical resonators with a geometry specified in Fig. REF (c). The geometry was chosen to accommodate the microphone and the speaker used for the measurements (see Methods), as well as to ensure a good separation of the discrete resonant modes, in the frequency domain we are interested in. The latter ensures that only basic mode-to-mode couplings occur. Figure REF (c) also shows the domain of the acoustic wave propagation and, as one can see, the resonators are connected through a thin domain, which we call the spacer. It is highlighted in red in Fig. REF (a). By filling this spacer with solid material, we can confine the wave propagation and create a re-configurable boundary, highlighted in red in Fig. REF (b), to control the phason as explained below. It is important to acknowledge that the use of a spacer as a solution for resonator coupling, and not other complicated means such as bridges, is one of the keys to our results. The spacer, certainly, does not allow any fine tuning but it does enable strong coupling, hence it is of crucial importance that accessing the 4D Hall physics does not rely on fine tuning.[1]} Let us specify that, in order to tightly pack the resonators and increase the strength of the coupling, the nearest neighboring resonators have been placed on opposite sides of the spacer.
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